11. Kern E. Kenyon, Stability of Solitary Wave

Stability of Solitary Wave

Kern E. Kenyon

4632 North Lane, Del Mar, California 92014‐4134 U.S.A.

A physical method allows the following conclusions to be stated: at finite height the solitary wave of elevation is stable whereas the solitary wave of depression is unstable. These theoretical results are consistent with the classical laboratory observations made by Russell [J.S. Russell, Report of the Committee on Waves, Seventh Meeting of the British Association for the Advancement of Science (1837), p. 417; Report on Waves, Fourteenth Meeting of the British Association for the Advancement of Science (1845), p. 311]. In the frame of reference that moves with the wave the static and dynamic pressure differences between the crest and the undisturbed level for the solitary elevation and between the trough and undisturbed level for the depression are balanced along the surface streamline, and the balance is valid at finite wave heights. This physical technique is adapted from Einstein's [A. Einstein, Naturwissenschaften 4, 509 (1916)] deep‐water wave model and applied to a shallow‐water model of the solitary wave, where the central assumption is that the horizontal component of the fluid motion is independent of depth. The static pressure difference is stabilizing whereas the dynamic pressure difference is destabilizing. Let the static/dynamic pressure balance be initially in effect at a particular finite height but then the wave height is increased or decreased a small amount by some wave growth or decay mechanism. After the wave‐height change the static/dynamic balance will no longer hold and the wave will be stable when the static/dynamic pressure difference tries to restore the wave height to its original value; otherwise the wave will be unstable.